The measurement method used in this study is related to that of (CM), but instead of the sun uses an artificial light source placed at a finite distance from the water surface, similar to the early work of . A sketch of the measurement principle is shown in Fig. . A camera and a light source are placed at virtually the same location. The camera is observing reflections of the light source on the water surface. Due to the fully specular, i.e. mirror-like, nature of reflections at the water surface, for every light ray, the reflected angle (γ) equals the incident angle (β). For a setup where the emitter (light source) and detector (camera) are placed at the same location, light is only detected if the surface normal of the water surface points towards the position of emitter and detector. Thus, whenever a speckle (a distorted, reflected image of the light source) is visible at a location u in the image, the wave slope must be s=tan⁡α=u/f, where α is the angle between the surface normal and the vertical and f is the focal length of the camera lens, see inset in Fig. . In two dimensions, this becomes s=sxsy=1fuv, with the image coordinates u and v. Thus, for every speckle that is visible in the image, the slope of the water surface at the point of reflection is known – independent of the distance to the water surface. By averaging many snapshots of reflections from the water surface, the probability of slopes can be determined from the probability of the occurrence of speckles at certain positions in the images.

A major difference between the CM method and the principle used in this study is the nature of the statistical averaging process. CM measured from an airplane flying at an altitude of about 600 m 2000 ft, . In our study, the instrument was placed at a distance of 8 m to the water surface on the bow of research vessel R/V Meteor. While in the photographs of CM, many water surface facets contributed to the signal at each point in the image, our images resolve speckles coming from the smallest ripples. Thus, while CM derived a PDF from a single photograph, implicitly assuming homogeneous conditions over a larger area on the Ocean surface, our method relies on conditions to be stationary for a certain time (typically 5–30 min) at a small footprint of a few square meters. As will be discussed later, an open question is how representative the conditions at the measurement spot are when larger scales are of interest.

Figure  shows the instrument that was built for the measurement of wave statistics during the R/V Meteor M91 cruise. A Basler acA2500-14gm monochrome gigabit ethernet camera with 2592×1944 pixels and a maximum frame rate of 14 Hz is placed at the center of an LED array built from 182 OSRAM SFH 4715S high power infrared (850 nm) emitters. To collimate the light, Carclo 10 mm Medium Spot Frosted lenses (#10413) are placed on each LED, decreasing their half angle to ±25∘. The lens used (Schneider Kreuznach Cinegon 1.4/8) had a focal length of 8.2 mm that gives a field of view of 38∘×30∘. A near-infrared band-pass filter, matched to the emission spectrum of the LEDs, is used to suppress ambient light.

Camera and light source are integrated into the Reflective Stereo Slope Gauge with its two line shaped light sources and cameras. The RSSG was also used in this study and is capable of measuring wave heights by stereo triangulation, as well as the probability of zero surface slope, from which a surface roughness parameter χ which is proportional to the mean square slope can be estimated .

During M91, the instruments were installed at the bow of R/V Meteor (see Fig. ). The cameras were facing down to the water surface. To avoid interference of the waves with the research vessel, the vessel was moving forward and against the direction of the wind at a speed of 1–2 kn during measurements. In the derivation of the method in Eq. (), it was tacitly assumed that the camera is looking down vertically, with its image plane parallel to the mean water level. On a moving ship, this is not the case at a given moment. The ship's pitch and roll need to be measured and accounted for in data processing. Ship motion was recorded with a Crossbow VG400 inertial measurement unit (IMU) that was located next to the RSSG at the bow of R/V Meteor.

The wave slope statistics instrument was installed at the bow of the German research vessel R/V Meteor during the M91 cruise in December 2012. The cruise explored the coastal upwelling regions off Peru. Both its departure and final destination were the port of Callao, Peru. During the cruise, wave slope statistics were collected whenever the micro-structure probe was deployed and the ship was moving at a slow pace of 1–2 kn upwind. This guaranteed minimal interaction between the waves in the footprint of the camera and the ship itself. During casts with the CTD rosette, however, Meteor's 360∘ bow thruster was used to hold the position and interfered massively with water flow and waves around the bow of the ship, making measurements impossible.

Measurements are presented from a total of 30 stations. Slope probability distributions are available for all night-time stations, as daylight affected the measurements. However, the most important statistical parameter, the mean square slope σs2 was determined for all stations with the help of the RSSG. The measurements of the RSSG are not affected by daylight, as it uses a spectral band around 950 nm, in which light is effectively absorbed by water vapour in the atmosphere. Only in conditions when sun glint is visible in the RSSG images can the data not be processed. The RSSG measures the probability for slope zero, its inverse, the surface roughness parameter χ is proportional to mss . The proportionality constant is calibrated against the night-time measurements of the probability density functions (see Appendix ). An overview of the stations for which processed measurements are available is given in Table .

Figure  shows all available mean square slope (mss) measurements from the M91 cruise. Green diamonds are mss values extracted from slope PDFs, where mss is obtained as a parameter of the fit of the model function given in Eq. (). Blue circles are additional measurements with the RSSG, where the mss is estimated from the probability of slope zero. The solid and dashed black lines represent the Cox and Munk (CM) parameterizations of mss with wind speed for clean water and surface slicks, respectively . The error of the mss measurements from the PDF method (diamond symbols) was estimated to be less than 4 % from laboratory data under controlled conditions .

Most of the measurements fall in between the two parameterizations. A notable exception are the open green symbols. These are measurements from a single station (#1775-2, see Table ) under very complex conditions. Strong currents, large swell and high wind speeds, all from different directions likely led to disturbances of the measurement by the vessel itself. Neglecting those data points, the CM clean parameterization is found to overestimate mss.

The scatter in the data set is not surprising, in fact, compared to results from other more recent studies , the correlation of mss and wind is relatively good. Nevertheless, it is obvious that there is no simple relationship between surface roughness (mss) and wind speed. Particularly in the upwelling regions in the Peruvian coastal waters, high biological activity leads to a strong production of surface active material. It is well known from many laboratory and field studies that these surface films inhibit small-scale waves and reduce surface roughness .

The missing direct link between wind speed and mss also has consequences for air-sea interaction studies. Transfer velocities have been found to scale better with mss than with wind speed, especially for the range of wind speeds observed during M91 and under the influence of surfactants e.g.. Thus, one consequence of the observed ambiguity of the mss to wind speed relation is that using wind speed only parameterizations to estimate (local) transfer velocities will cause large errors. These transfer velocities are required to estimate trace gas fluxes across the water surface from the measurement of air-water concentration differences. Therefore, the large errors associated with the transfer velocities propagate into large errors in the computed fluxes.

Waves typically have higher slopes in the alongwind than in the crosswind direction. The anisotropy parameter γ=σa2/σc2 is the ratio between the mss in the alongwind direction, σa2, and the crosswind mss (σc2). report γ to lie between 1.0 and 0.62. give relationships of the mss components with wind speed in their much larger dataset of 8 million globaly distributed satellite images, from which computes a functional dependence of γ on the inverse wind speed γ=0.585+0.76U10-1.

Despite the large scatter in the data, Fig.  shows that our dataset does not agree with this parameterization. No clear trend with wind speed is visible, the mean value is γmean=0.82. CM speculate that increased anisotropy (lower γ) should be the consequence of steady blowing winds, while gustiness should lead to more isotropic wave fields. During M91, wind conditions were usually relatively stable. Wind speed data is only available as one-minute mean values, so that gustiness is hard to estimate. The data set was classified into two subsets of similar size. Open circles represent measurements during which the standard deviation of wind speed was less than 4 % of the mean, solid circles measurements were variability was greater. No clear trend between the two groups is visible. The overall relatively high values of γ are likely also due to the fact that we encountered mostly rather mature seas, as it is well known, e.g. from our own laboratory validation experiment that young, growing waves have larger anisotropy (see Appendix A).

Only a weak asymmetry between the alongwind and crosswind mss is reported also by . They find two possible explanations for the discrepancy between their data and the CM results. One, they argue that CM were measuring waves that were not in equilibrium with the wind but still growing, leading to higher asymmetry. And two, the spatial resolution of their (satellite) data set is only 0.25∘×0.25∘ and the variability of wind directions over this large area may smear out the asymmetry. In the data presented here, the spatial resolution is much higher than that of any of the other studies, but due to the temporal averaging that is necessary to measure the slope PDF, variability of the wind speed and direction could have an impact. As was noted above, wind speeds and directions were quite constant during the M91 cruise and their variability are not likely to be the source for a low asymmetry. The other consequence of the conditions encountered with steady winds at moderate wind speeds is that waves were likely in equilibrium with the wind for most of the measurements. Thus, the data set of M91 is certainly not universally valid. Still, the disagreement with other data sets underlines that many factors affect the slope PDF and simple relations may not be able to capture all the complex interactions.

During the M91 cruise, R/V Meteor probed the Peruvian coastal waters in multiple transects orthogonal to the coast line. Figure  shows the geographic location of all the stations for which wave slope data was processed. The color code of the data points represents the fraction ϵ=σs2/σs,CM2, where σs2 is the average mss of all measurements at a particular station and σs,CM2 is the mss expected from the clean water CM parameterization for the average wind speed at the station. Thus, the lower the value of ϵ, the more likely it is that a surface film is damping the waves to reduce surface roughness. The data set is sparse and no clear trend of reduced roughness in rather coastal waters compared to more open ocean conditions is visible, although it appears that there are two hot spots with low values of ϵ in the North (6∘ S) and center (between 10 and 12∘ S) of the study area.

It is expected that surface roughness is reduced particularly in upwelling areas, where biological activity is high. To investigate this, Fig.  shows ϵ over the water temperature, as upwelling areas have lower relative ocean temperatures. Values of ϵ are obviously lower on average for lower water temperatures, which correspond to upwelling zones. The notable outlier at about 18 ∘C with a value of 1.16, which is the dark red point close to the coast in Fig. , occurred at very low wind speeds (estimated U10=1.8ms-1). Under these conditions, the CM relation may not be dependable and influences such as temperature stratification may severely alter the wind profile between a height of 35 m, where wind speed was measured, and the water surface. Thus, it is quite possible that the wind speed at the surface corresponded to a different U10 wind speed. As mss is relatively small for this range of wind speeds, even a small absolute error has large consequences for ϵ. At higher water temperatures, the scatter in ϵ increases, but values as low as 0.6 occur throughout the temperature range encountered during M91. This demonstrates that reduced surface roughness is pronounced in upwelling zones, but not limited to these.

During M91, at some stations very inhomogeneous surface conditions were encountered. An example is shown in Fig. , where isolated patches of reduced surface roughness are clearly visible. Under these conditions, exchange processes are also expected to be highly inhomogeneous. This poses a challenge for all types of air-sea interaction measurements which average over a larger footprint, from direct covariance to dual tracers but also for estimating air-sea fluxes from the measurement of air-water concentration gradients . The exact footprint over which measurements using these methods are averaged is not easy to determine and homogeneity of conditions usually has to be assumed. Local measurements of transfer velocities are possible with the Active Controlled Flux Technique e.g., which has a well-defined footprint of about 1 m2. ACFT measurements were also performed during the M91 cruise using the same measurement footprint as the wave slope system. Results for these measurements will be the subject of a future publication. To estimate local transfer velocities, a certain amount of temporal averaging is required. Under conditions such as in Fig. , when the slick patches are moving relative to the ship, it is important to have time resolved measurements of surface roughness to allow to separate slick from non-slick conditions in the averaging process. Figure  shows an approach to achieve this. The mean brightness of speckles in RSSG images was computed and its change over a time span of 5 min is plotted. The raw signal is very variable, but the shown running mean has a clear trend. About 150 s into the measurement, the ship enters a slick patch, surface roughness suddenly decreases and speckle brightness increases by more than one order of magnitude. The drastic change in surface conditions is also well visible in the two example RSSG images shown in Fig. . In these images, speckles appear as dark spots on a white background, as the image gray scale was inverted for better visibility. The left image was acquired before the ship entered the slick patch (first green line in Fig. ) and a typical wavy surface is visible with many speckles coming from different surface facets. In the right image (taken at the time marked by the second line in Fig. ), the damping effect of the slick on the small-scale roughness of the water surface is clearly visible. The line-like shape of the light source (see Fig. ) dominates the shape of the speckles, meaning that the water surface is almost flat, like a slightly bent mirror.

The shown time series is just one of many examples of sudden changes of the surface roughness encountered during M91, other series show that spatial scales of surface film patches can be on the order of 100 m . On one hand, these rapidly changing conditions pose additional challenges to the estimation of air-sea interaction processes, but on the other hand, the results also show that changes can be tracked on very short time scales by suitable wave statistics measurements.